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Sahithyan's S1
Sahithyan's S1 — Mathematics

Differential Equations

Equations which are composed of an unknown function and its derivatives.

Types

Ordinary Differential Equations

When a differential equation involves one independent variable, and one or more dependent variables.

An example:

dydx=cos(x)\frac{\text{d}y}{\text{d}x} = \cos(x)

Partial Differential Equations

When a differential equation involves more than one independent variables, and more than one dependent variables.

yx=yx=cos(x)\frac{\partial y}{\partial x} = y_x = \cos(x)

Linear

A linear differential equation is a differential equation that is defined by a linear polynomial in the unknown function (dependant variable) and its derivatives, that is an equation of the form:

P0(x)y+P1(x)y+...+Pn(x)y(n)+Q(x)=0P_0 (x) y + P_1 (x) y' +\,...\,+ P_n (x) y^{(n)} + Q(x) = 0

Here:

  • P0,P1,,Pn,QP_0, P_1,\dots,P_n,Q are all differentiable functions of xx, doesn’t depend on yy
  • y(x)y(x) is the unknown function
  • y(n)y^{(n)} denotes the nnth derivative of yy

Nonlinear

Nonlinear differential equations are any equations that cannot be written in the above form. In particular, these include all equations that include:

  • yy and/or its derivatives raised to any power other than 11
  • nonlinear functions of yy or any of its derivative
  • any product or function of these

Properties of Differential Equations

Order

Highest order derivative.

Degree

Power of highest order derivative.

Initial Value Problem (IVP)

A differential equation along with appropriate number of initial conditions.

Initial condition(s) is/are required to determine which solution (out of the infinite number of solutions) is the suitable one for the given problem.

Picard’s Existence and Uniqueness Theorem

Consider the below IVP.

dydx=f(x,y)  ;  y(x0)=y0\frac{\text{d}y}{\text{d}x} = f(x,y) \;;\; y(x_0)=y_0

Suppose: DD is an open neighbourhood in R2\mathbb{R}^2 containing the point (x0,y0)(x_0,y_0).

If ff and fy\frac{\partial{f}}{\partial{y}} are continuous functions in DD, then the IVP has a unique solution in some closed interval containing x0x_0.